Course Title:                  Introduction to Complex Analysis

Course Code:                 MATH 3275

Level:                             3

Number of Credits:         3

Semester:                       1

Prerequisite(s):                MATH 2277or equivalent

COURSE RATIONALE:

It is a well-known fact that there is no real number whose square is –1. Mathematicians have developed a system of numbers, called complex numbers, in which the square root of –1 does exist. When the ideas of calculus are applied to functions of a complex variable a powerful and elegant theory emerges, known as complex analysis.

Complex analysis is a powerful tool for solving a wide array of applied problems. It is related to several traditional mathematical disciplines such as real analysis, differential equations, algebra and topology. Common applications include wave propagation phenomena such as those occurring in electrodynamics, optics, fluid mechanics and quantum mechanics, diffusion problems such as heat and contaminant diffusion, engineering tasks such as the computation of buoyancy and resistance of wings, the flows in turbines and the design of optimal car bodies, and signal processing and communication theory.

This course is designed to provide students with the necessary background in complex analysis for the pursuit of advanced theoretical work in pure or applied mathematics.

COURSE DESCRIPTION:

This course provides an introduction to the theory and application of complex variables and complex functions. The properties of elementary complex functions are outlined, and the concept of analyticity is developed in its entirety. The most fundamental theorems are stated, proved and utilized throughout. Particular emphasis is placed on the development of integral calculus in the complex plane. Practice problems will be incorporated throughout to provide concrete examples of how to apply the theory.

A sound knowledge of introductory Real Analysis is required. For this reason, Analysis I is listed as a course prerequisite.

LEARNING OUTCOMES:

By the end of the course, students will be able to:

• Prove that a complex function is continuous, differentiable or analytic at a point or given region of the complex plane.
• Identify and construct analytic functions using the Cauchy Riemann equations.
• Describe the properties of conformal mappings, and find basic Mobius transforms across complex planes.
• Manipulate elementary complex functions (exponentials, trigonometric, logarithmic and hyperbolic functions).
• State and prove Cauchy’s Fundamental Theorem, Cauchy’s Integral Formulae (for simply-connected and multiply-connected regions).
• Evaluate contour integrals via the Cauchy theorems and Cauchy’s integral formulae.
• State and prove Taylor’s Theorem and Laurent’s Theorem.
• Investigate the convergence of a complex sequence.
• Test a complex power series for absolute or uniform convergence.
• Provide the Taylor or Laurent series representation for a complex function in a given region.
• Classify isolated singular points and compute residues at poles.
• Utilize the Residue Theorem to evaluate improper real integrals.

COURSE CONTENT:

Review of complex numbers:

• Algebraic and geometric representation of complex numbers;
• Euler’s formula;
• Rational powers and roots of complex numbers;
• Regions in the complex plane.

Analytic functions:

• Limits, continuity and differentiability;
• Cauchy Riemann equations;
• Analytic and harmonic functions;
• Introduction to conformal mapping.

Elementary functions:

• The complex exponential function;
• Trigonometric and Hyperbolic functions and inverses;
• The complex logarithm – definition, properties, branches and branch cuts.

Complex Integration:

• Statement and proof of Cauchy’s Fundamental Theorem, Cauchy’s Integral Formulae (for simply-connected and multiply-connected regions);
• Evaluation of contour integrals via the Cauchy theorems and Cauchy’s integral formulae;
• Statement and proof of Taylor’s Theorem and Laurent’s Theorem.

Series:

• Convergence of sequences and series;
• Power series – absolute and uniform convergence, integration and differentiation;
• Taylor and Laurent series.

Residues and Poles:

• Isolated singular points, residues and the Residue Theorem;
• Classifying isolated singular points;
• Residues at poles;
• Evaluation of improper real integrals by contour integration around poles.

TEACHING METHODOLOGY:

This course will be delivered through a combination of informative lectures and participative tutorials. The total 39 contact hours are broken down as follows: 33 hours of lectures, 6 tutorials. Supporting course materials will be posted on myeLearning.

COURSE ASSESSMENT:

The course assessment will be broken into two components; a coursework component worth 50% and a final exam worth 50%.

• Two course work exams will be given, each worth 15% of the student’s final grade.
• Four assignments (practical questions based on the theory done during lectures) will be given throughout the semester, each worth 5%.
• The final exam will be two hours in length and will be worth 50% of the final grade.

COURSE CALENDER:

 Week Lecture Topics Assignments Tutorial 1 Introduction/Course Overview Review of complex numbers: Algebraic and geometric representation of complex numbers; Euler’s formula; Rational powers and roots of complex numbers; Regions in the complex plane. Analytic functions: Limits, continuity and differentiability; Cauchy Riemann equations. None None 2 Analytic functions: Analytic and harmonic functions. Tutorial #1 Assignment 1 given Tutorial #1 3 Analytic functions: Introduction to conformal mappings. None None 4 Analytic functions: Introduction to conformal mappings. Elementary functions: The complex exponential function; Trigonometric and Hyperbolic functions and inverses; The complex logarithm – definition, properties, branches and branch cuts. Tutorial #2 Assignment 1 due Assignment 2 given Tutorial #2 5 Complex Integration: Statement and proof of Cauchy’s Fundamental Theorem (for simply-connected and multiply-connected regions). Coursework Exam #1 (15%) None 6 Complex Integration: Statement and proof of Cauchy’s Integral Formulae (for simply-connected and multiply-connected regions). Evaluation of contour integrals via the Cauchy theorems and Cauchy’s integral formulae. Tutorial #3 Assignment 2 due Assignment 3 given Tutorial #3 7 Complex Integration: Evaluation of contour integrals via the Cauchy theorems and Cauchy’s integral formulae. Statement and proof of Taylor’s Theorem. None None 8 Complex Integration: Statement and proof of Laurent’s Theorem. Series: Convergence of sequences and series; Tutorial #4 Assignment 3 due Assignment 4 given Tutorial #4 9 Series: Power series – absolute and uniform convergence, integration and differentiation. Taylor and Laurent series. None None 10 Series: Taylor and Laurent series. Residues and Poles: Isolated singular points, residues and the Residue Theorem. Classifying isolated singular points; Residues at poles; Tutorial #5 Assignment #4 due Tutorial #5 11 Residues and Poles: Evaluation of improper real integrals by contour integration around poles. Coursework Exam #2 (15%) None 12 Residues and Poles: Evaluation of improper real integrals by contour integration around poles. Tutorial #6 None Tutorial #6 13 Revision None None

REFERENCE MATERIAL:

Books:

Prescribed

• James Ward Brown, Ruel V. Churchill. Complex Variables and Applications (Eighth Edition). McGraw-Hill College, 2008.

This book is pedagogically sound, comprehensively addresses all element of the syllabus, and provides useful case studies and examples.

Recommended

• Theory and Problems of Complex Variables, Murray R. Spiegel, Schaum’s Outline Series , McGraw-Hill, 2nd  Edition (May 20, 2009).
• Complex Variables: Introduction and Application, M. J. Ablowitz, A. S. Fokas, Cambridge Texts in Applied Mathematics (April 28, 2003).
• Complex Analysis with Applications, Richard A. Silverman, Dover Publications; 1st Edition (September 20, 2010).
• Complex Analysis, Brian Tom and Harold Ramkissoon (locally published).

Online Resources:

1. Support material will be made available via myelearning.
2. http://mathforum.org/library/topics/complex_a/ - The MathForum Internet Mathematics Library is a curated list of online resources for Complex Analysis, including online lecture notes, software, and practice problems. The site is maintained by the Goodwin College of Professional Studies at Drexel University.